I'm reading a book on Brownian Motion, and they define the hitting time as $$T_x = \inf\{t > 0 : B(t) = x \}$$
Later on they state that $B(T_x)=x$.
Why would they use inf instead of min? With inf, if we have infinite amount of crossings/hits at x in finite time, couldn't we have $B(T_x)\neq x$?
When they define $T_x$ they haven't yet proved that the minimum is attained so they define it as infimum. But then we can use continuity of paths to prove that the infimum is actually a minimum. The fact that $B(T_x)=x$ follows from continuity of Brownian paths and the fact that the paths attain the value $x$ at some time $t$.
